Loop-erased Random Walk Research Articles

Scaling limits of loop-erased Markov chains on resistance spaces via a partial loop-erasing procedure

We introduce partial loop-erasing operators. We show that by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we get a process equivalent to the chronological loop-erased Markov chain. As an application, we construct loop-erased random paths on bounded domains of resistance spaces as the weak limit of the loop erasure of the Markov chains on a sequence of finite sets approximating the space, and the limit is independent of the approximating sequences. The random paths we constructed are simple paths almost surely, and they can be viewed as the loop-erasure of the paths of the diffusion process. Finally, we show that the scaling limit of the loop-erased random walks on the Sierpiński carpet graphs exists, and is equivalent to the loop-erased random paths on the Sierpińksi carpet.

Loop-erased random walk branch of uniform spanning tree in topological polygons

We consider uniform spanning tree (UST) in topological polygons with 2N marked points on the boundary with alternating boundary conditions. In an earlier work by Liu-Peltola-Wu, the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE8. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE2. The conclusion is a generalization of an earlier work by Han-Liu-Wu where the authors derive the scaling limit of the LERW branch of UST when N=2. When N=2, the limiting law is SLE2(−1,−1;−1,−1). However, the limiting law is no longer in the family of SLE2(ρ) process as long as N≥3.

Unwrapped two-point functions on high-dimensional tori

We study unwrapped two-point functions for the Ising model, the self-avoiding walk (SAW) and a random-length loop-erased random walk on high-dimensional lattices with periodic boundary conditions. While the standard two-point functions of these models have been observed to display an anomalous plateau behaviour, the unwrapped two-point functions are shown to display standard mean-field behaviour. Moreover, we argue that the asymptotic behaviour of these unwrapped two-point functions on the torus can be understood in terms of the standard two-point function of a random-length random walk model on . A precise description is derived for the asymptotic behaviour of the latter. Finally, we consider a natural notion of the Ising walk length, and show numerically that the Ising and SAW walk lengths on high-dimensional tori show the same universal behaviour known for the SAW walk length on the complete graph.

The Hölder continuity of the scaling limit of three-dimensional loop-erased random walk

Let β be the growth exponent of the loop-erased random walk (LERW) in three dimensions. We prove that the scaling limit of 3D LERW is almost surely h-Hölder continuous for all h<1∕β, but not 1∕β-Hölder continuous.

The Loewner difference equation and convergence of loop-erased random walk

We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner difference equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.

Loop-erased partitioning of a graph: mean-field analysis

Given a weighted finite graph G, we consider a random partition of its vertex set induced by a measure on spanning rooted forests on G. The latter is a generalized parametric version of the classical Uniform Spanning Tree measure which can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter q>0. The related random trees—identifying the blocks of the partition—tend to cluster nodes visited by the random walk on time scale 1∕q. We explore the emerging macroscopic structure by analyzing two-point correlations, as a function of the tuning parameter q. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block. This interaction potential can be seen as an affinity measure for “densely connected nodes” and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structure. For such geometries we show phase-transitions in the behavior of the random partition as a function of the tuning parameter and the edge weights. Moreover, as a corollary of our main results, we infer the right scaling of the parameters that give rise to the emergence of “giant” blocks.

Loop-erased random walks associated with Markov processes

A new class of loop-erased random walks (LERW) on a finite set, defined as functionals from a Markov chain is presented. We propose a scheme in which, in contrast to the general settings of LERW, the loop-erasure is performed on a non-markovian sequence and moreover, not all loops are erased with necessity. We start with a special example of a random walk with loops, the number of which at every moment of time does not exceed a given fixed number. Further we consider loop-erased random walks, for which loops are erased at random moments of time that are hitting times for a Markov chain. The asymptotics of the normalized length of such loop-erased walks is established. We estimate also the speed of convergence of the normalized length of the loop-erased random walk on a finite group to the Rayleigh distribution.

Scaling limits of the three-dimensional uniform spanning tree and associated random walk

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/\beta$, where $\beta\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.

Convergence of loop-erased random walk in the natural parameterization

Loop-erased random walk (LERW) is one of the most well-studied critical lattice models. It is the self-avoiding random walk that one gets after erasing the loops from a simple random walk in order or, alternatively, by considering the branches in a spanning tree chosen uniformly at random. We prove that planar LERW parameterized by renormalized length converges in the lattice-size scaling limit to SLE2 parameterized by 5∕4-dimensional Minkowski content. In doing this, we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theorem, several of our results about LERW are of independent interest. We give, for example, two-point estimates, estimates on maximal content, and a “separation lemma” for two-sided LERW.

External diffusion-limited aggregation on a spanning-tree-weighted random planar map

Let M be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the M-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on M run for m steps is of order m2/d+om(1), where d is the metric ball volume growth exponent for M (which was shown to exist by Ding and Gwynne (Comm. Math. Phys. 374 (2020) 1877–1934). By known bounds for d, one has 0.55051⋯≤2/d≤0.563315…. Along the way, we also prove that loop-erased random walk (LERW) on M typically travels graph distance m2/d+om(1) in m units of time and that the graph-distance diameter of a finite spanning-tree-weighted random planar map with n edges, with or without boundary, is of order n1/d+on(1) except on an event with probability decaying faster than any negative power of n. Our proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity.

A reverse Aldous–Broder algorithm

The Aldous–Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree formed by all the first-entrance edges has the law of a uniform spanning tree. Here we show that the tree formed by all the last-exit edges also has the law of a uniform spanning tree. This answers a question of Tom Hayes and Cris Moore from 2010. The proof relies on a bijection that is related to the BEST theorem in graph theory. We also give other applications of our results, including new proofs of the reversibility of loop-erased random walk, of the Aldous–Broder algorithm itself, and of Wilson’s algorithm.

Criticality and conformality in the random dimer model.

In critical systems, the effect of a localized perturbation affects points that are arbitrarily far from the perturbation location. In this paper, we study the effect of localized perturbations on the solution of the random dimer problem in two dimensions. By means of an accurate numerical analysis, we show that a local perturbation of the optimal covering induces an excitation whose size is extensive with finite probability. We compute the fractal dimension of the excitations and scaling exponents. In particular, excitations in random dimer problems on nonbipartite lattices have the same statistical properties of domain walls in spin glass. Excitations produced in bipartite lattices, instead, are compatible with a loop-erased self-avoiding random walk process. In both cases, we find evidence of conformal invariance of the excitations that is compatible with SLE_{κ} with parameter κ depending on the bipartiteness of the underlying lattice only.

A Multi-parameter Family of Self-avoiding Walks on the Sierpiński Gasket

In this paper, we construct a multi-parameter family of self-avoiding walks on the Sierpiński gasket. It includes the branching model, the loop-erased random walk and the loop-erased self-repelling walk. We reproduce in a unified manner the proof of the existence of the continuum limit and the self-avoiding property of the limit processes. Our limit processes include not only all the processes obtained from the previously studied self-avoiding walk models, but also the ones that have not been constructed before. While the paths of limit processes appearing in the previous works were self-avoiding or filled the whole space, our family includes continuous processes whose path is self-intersecting but does not fill the whole space.

An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson

We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the dd-dimensional hypercubic lattice, at large scales this theory reduces to a scalar \phi^4ϕ4-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field \phi(x)\in \mathbb Cϕ(x)∈ℂ (standard formulation) or a nilpotent one satisfying \phi(x)^2 =0ϕ(x)2=0. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.

Loop-Erased Random Walk as a Spin System Observable

The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk.

Near-critical spanning forests and renormalization

We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state space of abstract graphs with real-valued edge weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in order to relate the law of the scaling limit of a critical model to a stationary distribution of such a renormalization/Markov process. When starting from any two-dimensional lattice with constant edge weights, the Markov process does indeed converge in law to this stationary distribution that corresponds to a scaling limit of UST with Poissonian deletions. The results of this paper heavily build on the convergence in distribution of branches of the UST to $\mathrm{SLE}_{2}$ (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the loop-erased random walk to the “natural parametrization” of the $\mathrm{SLE}_{2}$ (a recent result by Lawler and Viklund).

On the convergence of massive loop-erased random walks to massive SLE(2) curves

Following the strategy proposed by Makarov and Smirnov [24] in 2009 (see also [3, 2] for theoretical physics arguments), we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up to [24] appeared since then, we believe that such a treatment might be of interest to the community. We do not require any regularity of the limiting planar domain near its degenerate prime ends a and b except that (Ωδ,aδ,bδ) are assumed to be ‘close discrete approximations’ to (Ω,a,b) near a and b in the sense of [13].

Fractal dimension of critical curves in the O(n) -symmetric ϕ4 model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY , and Heisenberg models

We calculate the fractal dimension d_{f} of critical curves in the O(n)-symmetric (ϕ[over ⃗]^{2})^{2} theory in d=4-ε dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at n=-2, self-avoiding walks (n=0), Ising lines (n=1), and XY lines (n=2), in agreement with numerical simulations. It can be compared to the fractal dimension d_{f}^{tot} of all lines, i.e., backbone plus the surrounding loops, identical to d_{f}^{tot}=1/ν. The combination ϕ_{c}=d_{f}/d_{f}^{tot}=νd_{f} is the crossover exponent, describing a system with mass anisotropy. Introducing a self-consistent resummation procedure and combining it with analytic results in d=2 allows us to give improved estimates in d=3 for all relevant exponents at 6-loop order.

The infinite two-sided loop-erased random walk

The loop-erased random walk (LERW) in $ {\mathbb {Z}}^{d}, d \geq 2$, is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the “middle” of the path.

Boundary Correlations in Planar LERW and UST

We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple $\mathrm{SLE}_\kappa$ at $\kappa=2$.